Reflexivity is not modally definable, in the sense that there is no modal formula that can specify only the reflexivity of points within a given model/frame (e.g., due to the bisimulation of a model with only a reflexive point and the the model of natural numbers with the natural order -- they must both satisfy the same formulas at bisimialr points, but can't do that for a formula that is supposed to characterize the reflixivity of a point). Similarly irreflexivity in a model/frame is not modally definable. So, and to narrow down our focus on the reflexivity:
- Reflexivity of points within models (i.e., for the the points in a model that bear that property) is NOT modally definable.
But the class of reflexive models/frames (i.e., the models/frames whose every point is reflexive) is modally definable: the modal formula $p \rightarrow \Diamond p$ characterizes such a class. (however, the class of irreflexive frames is still not modally definable, because it's not closed under the formation of bounded mrophic images.). So, again, to narrow down our focus on the reflexivity:
- The class of reflexive models/frames IS modally definable
Now, my questions: am I getting this right? and if yes, how do I understand the quesion "Is reflexivity modally definable?", and finally, is there any insight behind this different levels of definability?